Solve the inequality: $\quad-157 \leq 5-9 x$
Answer (2)
Subtract 5 from both sides: − 162 ≤ − 9 x .
Divide both sides by -9 (and flip the inequality): 18 ≥ x .
Rewrite the inequality: x ≤ 18 .
The solution to the inequality is: x ≤ 18
Explanation
Understanding the Inequality We are given the inequality − 157 ≤ 5 − 9 x . Our goal is to isolate x to solve for its possible values.
Subtracting 5 from Both Sides First, we subtract 5 from both sides of the inequality to start isolating the term with x : − 157 − 5 ≤ 5 − 9 x − 5
− 162 ≤ − 9 x
Dividing by -9 and Flipping the Inequality Next, we divide both sides of the inequality by -9. Remember that when we divide or multiply an inequality by a negative number, we must reverse the direction of the inequality sign: − 9 − 162 ≥ − 9 − 9 x
18 ≥ x
Rewriting the Inequality Finally, we can rewrite the inequality to express x in terms of 18: x ≤ 18
This means that x can be any value less than or equal to 18.
Examples
Understanding inequalities is crucial in many real-world scenarios. For instance, imagine you're managing a budget and need to ensure your expenses ( x ) don't exceed a certain limit (e.g., 18 ) . T h e in e q u a l i t y x \leq 18$ helps you visualize and maintain your spending within the allowable amount. Similarly, in manufacturing, quality control often involves ensuring that a product's dimensions fall within a specified range to meet standards. Inequalities are also used in optimization problems to find the best possible solution under given constraints, such as maximizing profit while minimizing costs.
To solve the inequality − 157 ≤ 5 − 9 x , we first subtract 5, giving us − 162 ≤ − 9 x . Then, dividing by − 9 (and reversing the inequality) results in x ≤ 18 . Therefore, the solution to the inequality is that x can be any value less than or equal to 18.
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